A Solution to the Arf-Kervaire Invariant Problem

Proceedings of 17th G¨okova Geometry-Topology Conference pp. 21 – 63

A solution to the Arf-Kervaire invariant problem

Michael A. Hill, Michael J. Hopkins and Douglas C. Ravenel

Abstract. This paper gives the history and background of one of the oldest problems in algebraic topology, along with an outline of our solution to it. A rigorous account can be found in our preprint [HHR]. The third author has a website with numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009.

Drawing by Carolyn Snaith, 1981

Partially supported by DARPA and NSF.

21 HILL, HOPKINS and RAVENEL

Contents 1. Background and history 24 1.1. Pontryagin’s early work on homotopy groups of spheres 24 1.2. Our main result 27 1.3. The manifold formulation 28 1.4. The unstable formulation 30 1.5. Questions raised by our theorem 32 2. Our strategy 33 2.1. Ingredients of the proof 33 2.2. The spectrum Ω 33 2.3. How we construct Ω 34 2.4. The norm 34 2.5. Fixed points and homotopy fixed points 35 2.6. RO(G)-graded homotopy 36 3. The slice filtration 37 3.1. Postnikov towers 37 3.2. Slice cells 38 3.3. The slice tower 38 3.4. The Slice Theorem 39 u (G) 3.5. Refining π∗ MU 40 3.6. The definition of ΩO 40 4. The Gap Theorem 41 4.1. Derivation from the Slice Theorem 41 4.2. An easy calculation 41 5. The Periodicity Theorem 44 5.1. Geometric fixed points 44 5.2. Some slice differentials 46 5.3. Some RO(G)-graded calculations 47 5.4. An even trickier RO(G)-graded calculation 47 5.5. The proof of the Periodicity Theorem 48 6. The Homotopy Fixed Point Theorem 49 7. The Detection Theorem 50 7.1. θj intheAdams-Novikovspectralsequence 50 7.2. Formal A-modules 52 7.3. The relation between MU (C8) and formal A-modules 53 7.4. DerivingtheDetectionTheoremfromLemma7.1 54 7.5. The proof of Lemma 7.1 54 8. The Slice and Reduction theorems 55 8.1. DerivingtheSliceTheoremfromtheReductionTheorem 56 8.2. The proof of the Reduction Theorem 57 References 61

22 A solution to the Arf-Kervaire invariant problem

S Main Theorem. The Arf-Kervaire elements θj ∈ π2j+1−2 do not exist for j ≥ 7.

S Here πk denotes the kth stable homotopy group of spheres, which will be defined shortly. The kth (for a positive integer k) homotopy group of the topological space X, denoted k by πk(X), is the set of continuous maps to X from the k-sphere S , up to continuous deformation. For technical reasons we require that each map send a specified point in Sk (called a base point) to a specified point x0 ∈ X. When X is path connected the choice of these two points is irrelevant, so it is usually omitted from the notation. When X is not path connected, we get different collections of maps depending on the path connected component of the base point. This set has a natural group structure, which is abelian for k > 1. The word natural here means that a continuous base point preserving map f : X → Y induces a homomorphism f∗ : πkX → πkY , sometimes denoted by πkf. n It is known that the group πn+kS is independent of n for n>k. There is a homomor- n n+1 n+1 n+k+1 phism E : πn+kS → πn+k+1S defined as follows. S [respectively S ] can be obtained from Sn [Sn+k] by a double cone construction known as suspension. The cone over Sn is an (n+1)-dimensional ball, and gluing two such balls together along their common boundary gives an (n+1)-dimensional sphere. A map f : Sn+k → Sn can be canonically extended (by suspending both its source and target) to a map Ef : Sn+k+1 → Sn+1, and this leads to the suspension homomorphism E. The Freudenthal Suspension Theo- rem [Fre38], proved in 1938, says that it is onto for k = n and an isomorphism for n>k. n For this reason the group πn+kS is said to be stable when n>k, and it is denoted by S πk and called the stable k-stem. j+1 The Main Theorem above concerns the case k = 2 − 2. The θj in the theorem is a hypothetical element related to a geometric invariant of certain manifolds studied originally by Pontryagin starting in the 1930s, [Pon38], [Pon50] and [Pon55]. The problem came into its present form with a theorem of Browder [Bro69] published in 1969. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved, namely that all of the θj exist. The θj in the theorem is the name given to a hypothetical map between spheres for which the Arf-Kervaire invariant is nontrivial. Browder’s theorem says that such things can exist only in dimensions that are 2 less than a power of 2. Some homotopy theorists, most notably Mahowald, speculated about what would happen if θj existed for all j. They derived numerous consequences about homotopy groups of spheres. The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis. After 1980, the problem faded into the background because it was thought to be too hard. In 2009, just a few weeks before we announced our theorem, Snaith published a book [Sna09] on the problem “to stem the tide of oblivion.” On the difficulty of the problem, he wrote

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In the light of ...the failure over fifty years to construct framed manifolds of Arf-Kervaire invariant one this might turn out to be a book about things which do not exist. This [is] why the quotations which preface each chapter contain a preponderance of utterances from the pen of Lewis Carroll. Our proof is two giant steps away from anything that was attempted in the 70s. We now know that the world of homotopy theory is very different from what they had envisioned then. 1. Background and history 1.1. Pontryagin’s early work on homotopy groups of spheres The Arf-Kervaire invariant problem has its origins in Pontryagin’s early work on a geometric approach to the homotopy groups of spheres, [Pon38], [Pon50] and [Pon55]. Pontryagin’s approach to maps f : Sn+k → Sn is to assume that f is smooth and that the base point y0 of the target is a regular value. (Any continuous f can be continuously −1 deformed to a map with this property.) This means that f (y0) is a closed smooth n+k n k-manifold M in S . Let D be the closure of an open ball around y0. If it is sufficiently small, then V n+k = f −1(Dn) in Sn+k is an (n + k)-manifold homeomorphic to M × Dn with boundary homeomorphic to M × Sn−1. It is also a tubular neighborhood of M k and comes equipped with a map p : V n+k → M k sending each point to the nearest point in M. For each x ∈ M, p−1(x) is homeomorphic to a closed n-ball Bn. The pair (p,f|V n+k) defines an explicit homeomorphism

(p,f|V n+k) n+k k n V ≈ / M × D .

This structure on M k is called a framing, and M is said to be framed in Rn+k. A choice n of basis of the tangent space at y0 ∈ S pulls back to a set of linearly independent normal vector ﬁelds on M ⊂ Rn+k. Conversely, suppose we have a closed sub-k-manifold M ⊂ Rn+k with a closed tubular neighborhood V and a homeomorphism h to M × Dn as above. This is called a framed sub-k-manifold of Rn+k. Some remarks are in order here. • The existence of a framing puts some restrictions on the topology of M. All of its characteristic classes must vanish. In particular it must be orientable. • A framing can be twisted by a map g : M → SO(n), where SO(n) denotes the group of orthogonal n × n matrices with determinant 1. Such matrices act on Dn in an obvious way. The twisted framing is the composite

V h / M k × Dn / M k × Dn

(m,x) / (m,g(m)(x)). We will say more about this later.

24 A solution to the Arf-Kervaire invariant problem

• If we drop the assumption that M is framed, then the tubular neighborhood V is a (possibly nontrivial) disk bundle over M. The map M → y0 needs to be replaced by a map to the classifying space for such bundles, BO(n). This leads to unoriented bordism theory, which was analyzed by Thom in [Tho54]. Two helpful references for this material are the books by Milnor-Stasheﬀ [MS74] and Stong [Sto68]. Pontryagin constructs a map P (M,h) : Sn+k → Sn as follows. We regard Sn+k as the one point compactiﬁcation of Rn+k and Sn as the quotient Dn/∂Dn. This leads to a diagram

p (V,∂V ) h / M × (Dn,∂Dn) 2 / (Dn,∂Dn) _

P (M,h) (Rn+k, Rn+k − intV ) / (Sn+k,Sn+k − intV ) / (Sn, {∞})

P (M,h) Sn+k − intV / {∞}

The map P (M,h) is the extension of p2h obtained by sending the complement of V in Sn+k to the point at inﬁnity in Sn. For n>k, the choice of the embedding (but not the choice of framing) of M into the Euclidean space is irrelevant. Any two embeddings (with suitably chosen framings) lead to the same map P (M,h) up to continuous deformation. To proceed further, we need to be more precise about what we mean by continuous deformation. Two maps f1,f2 : X → Y are homotopic if there is a continuous map h : X × [0, 1] → Y (called a homotopy between f1 and f2) such that

h(x, 0) = f1(x) and h(x, 1) = f2(x). n+k n Now suppose X = S , Y = S , and the map h (and hence f1 and f2) is smooth with −1 y0 as a regular value. Then h (y0) is a framed (k +1)-manifold N whose boundary is the −1 −1 disjoint union of M1 = f1 (y0) and M2 = f2 (y0). This N is called a framed cobordism between M1 and M2, and when it exists the two closed manifolds are said to be framed cobordant. fr n+k Let Ωk,n denote the cobordism group of framed k-manifolds in R . The above construction leads to Pontryagin’s isomorphism ≈ fr n Ωk,n / πn+kS .

First consider the case k = 0. Here the 0-dimensional manifold M is a ﬁnite set of points in Rn. Each comes with a framing which can be obtained from a standard one by an element in the orthogonal group O(n). We attach a sign to each point corresponding to the sign of the associated determinant. With these signs we can count the points algebraically and get an integer called the degree of f. Two framed 0-manifolds are cobordant iﬀ they have the same degree.

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Sidebar 1 The Hopf-Whitehead J-homomorphism

Suppose our framed manifold is Sk with a framing that extends to a Dk+1. This will n lead to the trivial element in πn+k(S ), but twisting the framing can lead to nontrivial elements. The twist is determined up to homotopy by an element in πk(SO(n)). Pontryagin’s construction thus leads to the homomorphism

J n πkSO(n) / πn+kS introduced by Hopf [Hop35] and Whitehead [Whi42]. Both source and target are known to be independent of n for n>k +1. In this case the source group for each k (denoted simply by πkSO since n is irrelevant) was determined by Bott [Bot59] in his remarkable periodicity theorem. He showed Z for k ≡ 3 or 7 mod 8 πkSO =  Z/2 for k ≡ 0 or 1 mod 8  0 otherwise. Here is a table showing these groups for k ≤ 10. k 1 2 3 4 5 6 7 8 9 10 πk(SO) Z/2 0 Z 0 0 0 Z Z/2 Z/2 0 In each case where the group is nontrivial, the image under J of its generator is known to generate a direct summand. In the jth case we denote this image by βj and its dimension by φ(j), which is roughly 2j. The ﬁrst three of these are the Hopf maps S S S S S S η ∈ π1 , ν ∈ π3 and σ ∈ π7 . After that we have β4 ∈ π8 , β5 ∈ π9 , β6 ∈ π11 and so on. For the case π4m−1SO = Z, the image under J is known to be a cyclic group whose order am is the denominator of Bm/4m, where Bm is the mth Bernoulli number. Details can be found in [Ada66] and [MS74]. Here is a table showing these values for m ≤ 10. m 1 2 3 4 5 6 7 8 9 10 am 24 240 504 480 264 65,520 24 16,320 28,728 13,200

Now consider the case k = 1. M is a closed 1-manifold, i.e., a disjoint union of circles. Two framings on a single circle diﬀer by a map from S1 to the group SO(n), and it is known that

0 for n = 1 π1SO(n)=  Z for n = 2  Z/2 for n> 2.  26 A solution to the Arf-Kervaire invariant problem

It turns out that any disjoint union of framed circles is cobordant to a single framed circle. This can be used to show that 0 for n = 1 n πn+1S =  Z for n = 2  Z/2 for n> 2.  The case k = 2 is more subtle. As in the 1-dimensional case we have a complete classification of closed 2-manifolds, and it is only necessary to consider ones which are path connected and oriented. The existence of a framing implies that the surface is orientable, so it is characterized by its genus. If the genus is zero, namely if M = S2, then there is a framing which extends to a 3-dimensional ball. This makes M cobordant to the empty set, which means that the map is null homotopic (or, more briefly, null), meaning that it is homotopic to a constant 2 map. Any two framings on S differ by an element in π2SO(n). This group is known to 2 vanish (as is π2 of any Lie group), so any two framings on S are equivalent, and the map f : Sn+2 → Sn is null. Now suppose the genus is one and suppose we can find an embedded arc on which the framing extends to a disk in Sn+2. Then there is a cobordism which effectively cuts along the arc, leaving a surface bounded by two circles, and attaches two framed disks to them. This process is called framed surgery. If we can do this, then we have converted the torus to a 2-sphere and shown that the map f : Sn+2 → Sn is null. When can we find such a closed curve in M? It must represent a generator of H1(M) and carry a trivial framing. This leads to a map

ϕ : H1(M; Z/2) → Z/2 (1) deﬁned as follows. Each class in H1 can be represented by a closed curve which is framed either trivially or nontrivially. It can be shown that homologous curves have the same framing invariant, so ϕ is well deﬁned. At this point Pontryagin made a famous mistake which went undetected for over a decade: he assumed that ϕ was a homomorphism. We now know this is not the case, and we will say more about it below in §1.3. On that basis he argued that ϕ must have a nontrivial kernel, since the source group is (Z/2)2. Therefore there is a closed curve along which we can do the surgery. It follows that n M can be surgered into a 2-sphere, leading to the erroneous conclusion that πn+2(S ) = 0 for all n. Freudenthal [Fre38] and later George Whitehead [Whi50] both proved that it is Z/2 for n ≥ 2. Pontryagin corrected his mistake in [Pon50], and in [Pon55] he gave a complete account of the relation between framed cobordism and homotopy groups of spheres.

1.2. Our main result Our main theorem can be stated in three diﬀerent but equivalent ways:

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• Manifold formulation: It says that a certain geometrically defined invariant Φ(M) (the Arf-Kervaire invariant, to be defined later) on certain manifolds M is always zero. • Stable homotopy theoretic formulation: It says that certain long sought hypothetical maps between high dimensional spheres do not exist. • Unstable homotopy theoretic formulation: It says something about the EHP sequence (to be defined below), which has to do with unstable homotopy groups of spheres. Here again is the stable homotopy theoretic formulation.

S Main Theorem. The Arf-Kervaire elements θj ∈ π2j+1−2 do not exist for j ≥ 7. 1.3. The manifold formulation

Let λ be a nonsingular anti-symmetric bilinear form on a free abelian group H of rank 2n with mod 2 reduction H. It is known that H has a basis of the form {ai,bi : 1 ≤ i ≤ n} with

λ(ai,ai′ ) = 0 λ(bj,bj′ )=0 and λ(ai,bj )= δi,j . In other words, H has a basis for which the bilinear form’s matrix has the symplectic form 0 1  1 0  0 1    1 0    .  .   ..     0 1     1 0    A quadratic reﬁnement of λ is a map q : H → Z/2 satisfying q(x + y)= q(x)+ q(y)+ λ(x,y). Its Arf invariant is n

Arf(q)= q(ai)q(bi) ∈ Z/2. Xi=1 In 1941 Arf [Arf41] proved that this invariant (along with the number n) determines the isomorphism type of q. An equivalent deﬁnition is the “democratic invariant” of Browder [Bro69]. The elements of H “vote” for either 0 or 1 by the function q. The winner of the election (which is never a tie) is Arf(q). Here is a table illustrating this for three possible reﬁnements q, q′ and q′′ when H has rank 2.

28 A solution to the Arf-Kervaire invariant problem

x 0 a b a + b Arf invariant q(x) 000 1 0 q′(x) 011 1 1 q′′(x) 010 0 0 The value each reﬁnement on a+b is determined by those on a and b, and q′′ is isomorphic to q. Thus the vote is three to one in each case. When H has rank 4, it is 10 to 6.

Let M bea2m-connected smooth closed manifold of dimension 4m +2 with a framed embedding in R4m+2+n. We saw above that this leads to a map f : Sn+4m+2 → Sn and n hence an element in πn+4m+2S . Let H = H2m+1(M; Z), the homology group in the middle dimension. Each x ∈ H is 2m+1 represented by an immersion ix : S # M with a stably trivialized normal bundle. H has an antisymmetric bilinear form λ defined in terms of intersection numbers. In 1960 Kervaire [Ker60] defined a quadratic refinement q on its mod 2 reduction in terms of the trivialization of each sphere’s normal bundle. The Kervaire invariant Φ(M) is defined to be the Arf invariant of q. In the case m = 0, when the dimension of the manifold is 2, Kervaire’s q is Pontryagin’s map ϕ of (1). What can we say about Φ(M)? • Kervaire [Ker60] showed it must vanish when k = 2. This enabled him to construct the first example of a topological manifold (of dimension 10) without a smooth structure. Let N be a smooth 10-manifold with boundary given as the union of two copies of the tangent disk bundle of S5, glued together along a common copy of D5 × D5 where the fibers in one copy are parallel to the base in the other. The boundary is homeomorphic to S9. Thus we can get a closed topological manifold X by gluing on a 10-ball along its common boundary with n, or equivalently collapsing ∂N to a point. X then has nontrivial Kervaire invariant. On the other hand, Kervaire proved that any smooth framed manifold must have trivial Kervaire invariant. Therefore the topological framed manifold X cannot have a smooth structure. Equivalently, the boundary ∂N cannot be diffeomorphic to S9. It must be an exotic 9-sphere. • For k = 0 there is a framing on the torus S1 × S1 ⊂ R4 with nontrivial Kervaire invariant. Pontryagin used it in [Pon50] (after some false starts in the 30s) to n show πn+2S = Z/2 for all n ≥ 2. • There are similar constructions for k = 1 and k = 3, where the framed manifolds are S3 × S3 and S7 × S7 respectively. Like S1, S3 and S7 are both parallelizable, meaning that their tangent bundles are trivial. The framings can be twisted in such a way as to yield a nontrivial Kervaire invariant. • Brown-Peterson [BP66b] showed that it vanishes for all positive even k. This means that apart from the 2-dimensional case, any smooth framed manifold with nontrivial Kervaire invariant must have a dimension congruent to 6 modulo 8. • Browder [Bro69] showed that it can be nontrivial only if k = 2j−1 − 1 for some 2 positive integer j. This happens iff the element hj is a permanent cycle in the

29 HILL, HOPKINS and RAVENEL

Adams spectral sequence, which was originally introduced in [Ada58]. (More information about it can be found in [Rav86] and [Rav04].) The corresponding S element in πn+2j+1−2 is θj, the subject of our theorem. This is the stable homotopy theoretic formulation of the problem. • θj is known to exist for 1 ≤ j ≤ 3, i.e., in dimensions 2, 6, and 14. In these j j cases the relevant framed manifold is S2 −1 × S2 −1 with a twisted framing as 2j −1 discussed above. The framings on S represent the elements hj in the Adams spectral sequence. The Hopf invariant one theorem of Adams [Ada60] says that for j > 3, hj is not a permanent cycle in the Adams spectral sequence because it supports a nontrivial differential. (His original proof was not written in this language, but had to do with secondary cohomology operations.) This means that for j > 3, a smooth framed manifold representing θj (i.e., having a nontrivial j j Kervaire invariant) cannot have the form S2 −1 × S2 −1. • θj is also known to exist for j = 4 and j = 5, i.e., in dimensions 30 and 62. In both cases the existence was first established by purely homotopy theoretic means, without constructing a suitable framed manifold. For j = 4 this was done by Barratt, Mahowald and Tangora in [MT67] and [BMT70]. A framed 30-manifold with nontrivial Kervaire invariant was later constructed by Jones [Jon78]. For j = 5 the homotopy theory was done in 1985 by Barratt-Jones- Mahowald in [BJM84]. • Our theorem says θj does not exist for j ≥ 7. The case j = 6 is still open. Kervaire’s 10-manifold with boundary that is described above can be generalized to a (4k +2)-manifold with boundary constructed in a similar way. In all cases except k = 0, 1 or 3, any framing of this manifold will do because the tangent bundle of S2k+1 is nontrivial and leads to a nontrivial invariant. The boundary is homeomorphic to S4k+1, but may or may not be diffeomorphic to the standard sphere. If it is, then attaching a (4k + 2)-disk to it will produce a smooth framed manifold with nontrivial Kervaire invariant. If it is not, then we have an exotic (4k + 1)-sphere bounding a framed manifold and hence not detected by framed cobordism.

1.4. The unstable formulation

Assume all spaces in sight are localized at the prime 2. For each n> 0 there is a ﬁber sequence due to James, [Jam55], [Jam56a], [Jam56b] and [Jam57]

Sn E / ΩSn+1 H / ΩS2n+1. Here ΩX = Ω1X where ΩkX denotes the space of continuous base point preserving maps to X from the k-sphere Sk, known as the kth loop space of X. This leads to a long exact sequence of homotopy groups

n E n+1 H 2n+1 P n ... / πm+nS / πm+n+1S / πm+n+1S / πm+n−1S / ...

30 A solution to the Arf-Kervaire invariant problem

Here • E stands for Einh¨angung, the German word for suspension. • H stands for Hopf invariant. • P stands for Whitehead product. Assembling these for ﬁxed m and various n leads to a diagram

2n−1 2n+1 2n+3 πm+n+1S πm+n+2S πm+n+3S

P P P E n−1 E n E n+1 E ... / πm+n−1S / πm+nS / πm+n+1S / ...

H H H 2n−3 2n−1 2n+1 πm+n−1(S ) πm+nS πm+n+1S where • Sequences of arrows labeled H, P , E, H, P (or any subset thereof) in that order are exact. • The groups in the top and bottom rows are inductively known, and we can compute those in the middle row by induction on n. • The groups in the top and bottom rows vanish for large n, making E an isomorphism. • An element in the middle row has trivial suspension (is killed by E) iﬀ it is in the image of P . • It desuspends (is in the image of E) iﬀ its Hopf invariant (image under H) is trivial. When m = n − 1 this diagram is

n+1 π2n+1S

H 2n−1 2n+1 2n+3 π2nS π2n+1S = Z π2n+2S = 0

P P P E n−1 E n E n+1 E ... / π2n−2S / π2n−1S / π2nS / ...

H H H 2n−3 2n−1 2n+1 π2n−2S π2n−1S = Z π2nS = 0

2n+1 n The image under P of the generator of π2n+1S is denoted by wn ∈ π2n−1S and is called the Whitehead square.

• When n is even, H(wn)=2 and wn has inﬁnite order. 2n+1 • wn is trivial for n = 1, 3 and 7. In those cases the generator of π2n+1S is the n+1 Hopf invariant (image under H ) of one of the three Hopf maps in π2n+1S ,

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η S3 / S2, S7 ν / S4 and S15 σ / S8.

• For other odd values of n, twice the generator of the upper Z is H(wn+1), so wn has order 2. j+1 • It turns out that wn is divisible by 2 iﬀ n = 2 − 1 and θj exists, in which case wn = 2θj . 2n+1 • Each Whitehead square w2n+1 ∈ π4n+1S (except the cases n = 0, 1 and 3) desuspends to a lower sphere until we get an element with a nontrivial Hopf invariant, which is always some βj (see Sidebar 1, page 26). More precisely we have

H(w(2s+1)2j −1)= βj for each j > 0 and s ≥ 0. This result is essentially Adams’ 1962 solution to the vector ﬁeld problem [Ada62]. Recall the EHP sequence

n E n+1 H 2n+1 P n ... / πm+nS / πm+n+1S / πm+n+1S / πm+n−1S / ...

2n+1 Given some βj ∈ πφ(j)+2n+1S for φ(j) < 2n, one can ask about the Hopf invariant of its image under P , which vanishes when βj is in the image of H. In most cases the answer is known and is due to Mahowald, [Mah67] and [Mah82]. The remaining cases have to do with θj. The answer that he had hoped for is the following, which can be found in [Mah67]. (To our knowledge, Mahowald never referred to this as the World Without End Hypothesis. We chose that term to emphasize its contrast with the Doomsday Hypothesis.) World Without End Hypothesis (Mahowald 1967). S • The Arf-Kervaire element θj ∈ π2j+1−2 exists for all j > 0. 2j+1−1−φ(j) • It desuspends to S and its Hopf invariant is βj . • Let j,s > 0 and suppose that m = 2j+2(s + 1) − 4 − φ(j) and j+1 n = 2 (s + 1) − 2 − φ(j). Then P (βj ) has Hopf invariant θj. This describes the systematic behavior in the EHP sequence of elements related to the image of J, and the θj are an essential part of the picture. Because of our theorem, we now know that this hypothesis is incorrect. 1.5. Questions raised by our theorem

EHP sequence formulation. The World Without End Hypothesis was the nicest possible statement of its kind given all that was known prior to our theorem. Now we know it cannot be true since θj does not exist for j ≥ 7. This means the behavior of the indicated elements P (βj ) for j ≥ 7 is a mystery.

32 A solution to the Arf-Kervaire invariant problem

2 Adams spectral sequence formulation. We now know that the hj for j ≥ 7 are not permanent cycles, so they have to support nontrivial diﬀerentials. We have no idea what their targets are.

Manifold formulation. Here our result does not lead to any obvious new questions. It appears rather to be the ﬁnal page in the story.

Our method of proof offers a new tool for studying the stable homotopy groups of spheres. We look forward to learning more with it in the future. 2. Our strategy 2.1. Ingredients of the proof Our proof has several ingredients. • It uses methods of stable homotopy theory, which means it uses spectra instead of topological spaces. Recall that a space X has a homotopy group πkX for each positive integer k. A spectrum X has an abelian homotopy group πkX defined for 0 0 every integer k. For the sphere spectrum S , πkS is the stable k-stem homotopy S j+1 group πk . The hypothetical θj is an element of this group for k = 2 − 2. • It uses complex cobordism theory, the subject of Sidebar 2. This is a branch of algebraic topology having deep connections with algebraic geometry and number theory. It includes some highly developed computational techniques that began with work by Milnor [Mil60], Novikov ([Nov60], [Nov62] and [Nov67]) and Quillen [Qui69] in the 60s. A pivotal tool in the subject is the theory of formal group laws, the subject of Sidebar 4. • It also makes use of newer less familiar methods from equivariant stable homotopy theory. A helpful introduction to this subject is the paper of Greenlees-May [GM95]. This means there is a finite group G (a cyclic 2-group) acting on all spaces in sight, and all maps are required to commute with these actions. When we pass to spectra, we get homotopy groups indexed not just by the integers Z, but by RO(G), the orthogonal representation ring of G. Our calculations make use of this richer structure. 2.2. The spectrum Ω

We will produce a map S0 → Ω, where Ω is a nonconnective spectrum (meaning that it has nontrivial homotopy groups in arbitrarily large negative dimensions) with the following properties: (i) Detection Theorem. It has an Adams-Novikov spectral sequence (which is a device for calculating homotopy groups) in which the image of each θj is nontrivial. This means that if θj exists, we will see its image in π∗Ω. (ii) Periodicity Theorem. It is 256-periodic, meaning that πkΩ depends only on the reduction of k modulo 256.

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(iii) Gap Theorem. πkΩ=0 for −4

0 If θ7 ∈ π254S exists, (i) implies it has a nontrivial image in π254Ω. On the other hand, (ii) and (iii) imply that π254Ω = 0, so θ7 cannot exist. The argument for θj for larger j j+1 is similar, since |θj | = 2 − 2 ≡−2 mod 256 for j ≥ 7.

2.3. How we construct Ω The construction of Ω requires the use of equivariant stable homotopy theory. It is discussed extensively in [HHR, §2 and Appendices A and B]. Our spectrum Ω will be the fixed point spectrum for the action of C8 (the cyclic group of order 8) on an equivariant spectrum ΩO. To construct it we start with the complex cobordism spectrum MU. It can be thought of as the set of complex points of an algebraic variety defined over the real numbers. This means that it has an action of C2 defined by complex conjugation. The resulting C2-equivariant spectrum is denoted by MUR and is called real cobordism theory. This ter- minology follows Atiyah’s definition of real K-theory [Ati66], by which he meant complex K-theory equipped with complex conjugation.

2.4. The norm G Next we use a formal tool we call the norm NH for inducing up from an H-spectrum to a G-spectrum when H is a subgroup of G. It is treated in [HHR, §2.3]. Since the only G g groups we consider here are cyclic, we will denote NH by Nh where h = |H| and g = |G|. The analogous construction on the space level is easy to desribe. Let X be a space that is acted on by a subgroup H of G. Let

Y = MapH (G, X), the space of H-equivariant maps from G to X. Here the action of H on G is by left multiplication, and the resulting object has an action of G by right multiplication. As a set, Y = X|G/H|, the |G/H|-fold Cartesian power of X. A general element of G permutes these factors, each of which is left invariant by the subgroup H. For an H-spectrum X, G the analogous G-spectrum is denoted by NH X. In particular for a ﬁnite cyclic 2-group G = C2n+1 we get a G-spectrum

(G) g MU = N2 MUR

n underlain by MU (2 ). We are most interested in the case n = 2. This spectrum is not periodic, but it has a close relative ΩO (to deﬁned below in §5) which is periodic.

34 A solution to the Arf-Kervaire invariant problem

Sidebar 2 Complex cobordism

The complex cobordism spectum MU is a critical tool in modern stable homotopy theory. Its use in computing the stable homotopy groups of spheres is the subject of [Rav86]. Much of the needed background material on vector bundles and Thom spaces can be found in [MS74].

Let U(n) denote the n-dimensional unitary group and BU(n) its classifying space. The latter is the space of complex n-planes in an infinite dimensional complex vector space. It is the base space of an n-dimensional complex vector bundle where the fiber at each point is the set of vectors in the given n-plane. We denote the one point compactification of its total space (also known as the Thom space of the vector bundle) by MU(n). The inclusion map U(n) → U(n +1) leads to a map from Σ2MU(n) to MU(n +1), which is adjoint to the map MU(n) → Ω2MU(n + 1).

Such maps can be looped and iterated, allowing us to deﬁne 2n−k MUk = lim Ω MU(n) →

with MUk homeomorphic to ΩMUk+1. The collection {MUk} is the spectrum MU. Its homotopy groups are deﬁned by

πiMU = lim π2n+iMU(n)= πi+kMUk. → Its homology and cohomology can be similarly deﬁned. The inclusions of U(m) × U(n) into U(m + n) lead to maps MU(m) ∧ MU(n) → MU(m + n) and MU ∧ MU → MU. This makes MU into an E∞-ring spectrum. There are generalized homology and ∗ cohomology theories MU∗ and MU deﬁned by i MUi(X)= πiX ∧ MU and MU (X)=[X,MUi], with the latter being endowed with cup products.

We know that

π∗MU = Z[xi : i> 0] and H∗MU = Z[mi : i> 0]

where |xi| = |mi| = 2i. After localization at a prime p, MU splits into a wedge of copies of a spectrum BP (for Brown-Peterson [BP66a]) with

π∗BP = Z(p)[vi : i> 0] and H∗BP = Z(p)[ℓi : i> 0] i where |vi| = |ℓi| = 2(p − 1). This is related to a corresponding fact about formal groups laws described in Sidebar 4 (page 51). In this paper we will assume that MU has been localized at the prime 2.

2.5. Fixed points and homotopy ﬁxed points For a G-space X, XG is the subspace ﬁxed by all of G, which is the same as the space hG of equivariant maps from a point to X, MapG(∗,X). To get X , we replace the point 35 HILL, HOPKINS and RAVENEL

hG here by a free contractible G-space EG. The homotopy type of X = MapG(EG,X) is known to be independent of the choice of EG. The unique map EG → ∗ leads to a map ϕ : XG → XhG. We construct a C8-spectrum ΩO and show that

hC8 • ΩO satisﬁes the detection and Periodicity Theorems. C8 • ΩO satisﬁes the Gap Theorem. Hence our proof depends on a fourth property:

C8 hC8 (iv) Fixed Point Theorem. The map ΩO → ΩO is an equivalence. It proof is given in [HHR, §10]. This ﬁxed point set is our spectrum Ω.

We will come back to the deﬁnition of ΩO below.

2.6. RO(G)-graded homotopy Let RO(G) denote the orthogonal representation ring of G and let SV denote the one point compactiﬁcation of the orthogonal representation V . For a G-space or spectrum X deﬁne G V πV X =[S ,X]G. G Note that when the action of G on V is trivial and V has dimension |V |, we denote πV X G V dimV G by π|V |X, and an equivariant map S = S → X must land in X , so

G G πn X = πnX . In the stable category we can make sense of this for virtual as well as actual represen- tations, so we get homotopy groups indexed by RO(G), which we denote collectively by G u π⋆ X. We denote the ordinary homotopy of the underlying spectrum of X by π∗ X. We use the norm in two ways: • Externally as a functor from H-spectra to G-spectra. In particular, for a repre- g V indV sentation V of H, Nh S = S , the one point compactiﬁcation of the induced representation of V . • Internally as a homomorphism

g H G Nh : π⋆ X → π⋆ X for a G-ring spectrum X. Here we are using the forgetful functor to regard X as an H-spectrum with RO(H)-graded homotopy. The external norm takes us to G g g (g/h) π⋆ Nh X. Then Nh X is X as a G-spectrum, and we use the multiplication g on X to get a map Nh X → X. We will use the same notation for both. Recall that

π∗MU = Z(2)[x1,x2,... ] with |xi| = 2i.

36 A solution to the Arf-Kervaire invariant problem

(We are assuming that MU has been localized at 2.) It turns out that any choice of 2i generator xi : S → MU is the image of the forgetful functor of a map

xi Siρ MU/ R.

Here ρ denotes the regular real representation of C2, which is the same thing as the complex numbers C acted on by conjugation. u (G) For G = C2n+1 , π∗ MU is a graded polynomial algebra over Z(2) where • there are 2n generators in each positive even dimension 2i. • they are acted on transitively by G.

For a group generator γ ∈ G and polynomial generator ri ∈ π2i, the set

j n γ ri : 0 ≤ j < 2 2n i is algebraically independent, and γ ri =(−1) ri.

3. The slice ﬁltration Now we introduce our main technical tool, which is treated in [HHR, §6]. It is an equivariant analog of the Postnikov tower.

3.1. Postnikov towers First we need to recall some things about the classical Postnikov towers. The mth Postnikov section P mX of a space or spectrum X is obtained by killing all homotopy groups of X above dimension m by attaching cells. The fiber of the map X → P mX is Pm+1X, the m-connected cover of X. These two functors have some universal properties. Let S and S>m denote the cate- gories of spectra and m-connected spectra. m The functor P is Farjoun nullification [Far96] with respect to the subcategory S>m. This means the map X → P mX is universal among maps from X to spectra which are S>m-null in the sense that all maps to them from m-connected spectra are null. In other words, m • The spectrum P X is S>m-null. m • For any S>m-null spectrum Z, the map S(P X,Z) →S(X,Z) is an equivalence. m m−1 Since S>m ⊂ S>m−1, there is a natural transformation P → P , whose fiber is m denoted by Pm X. It is an Eilenberg-Mac Lane spectrum with homotopy concentrated in dimension m. In what follows G will be a finite cyclic 2-group, and g = |G|. Let SG denote the category of G-equivariant spectra. We need an equivariant analog of S>m. Our choice for this is somewhat novel.

37 HILL, HOPKINS and RAVENEL

3.2. Slice cells

Recall that S>m, the category of m-connected spectra, is the category of spectra built up out of spheres of dimension >m using arbitrary wedges and mapping cones. For a subgroup H of G with |H| = h and an integer k, let

kρH S(kρH )= G+ ∧H S where ρH denotes the regular realb representation of H. Its underlying spectrum is a wedge of g/h spheres of dimension kh which are permuted by elements of G and are invariant under H. We will replace the set of sphere spectra by the collection −1 A = S(kρH ), Σ S(kρH ): H ⊂ G, k ∈ Z . n o Definition 3.1. We will referb to the elementsb in the set A above as slice cells or simply −2 as cells. (Note that Σ S(kρH ) and larger desuspensions are not cells.) A free cell is one of the form S(kρ ), a wedge of g k-spheres permuted by G. Note that {e} b −1 b Σ S(kρ{e})= S((k − 1)ρ{e}). −1 Nonfree cells are said to be isotropicb . Cells ofb the form S(kρH ) (rather than Σ S(kρH )) are said to be pure. b b G In order to define S>m, we need to assign a dimension to each element in A. We do this in terms of the underlying wedge summands, namely −1 dim S(kρH )= kh and dimΣ S(kρH )= kh − 1. G Then S>m is the categoryb built up out of elementsb in A of dimension > m using arbitrary wedges, mapping cones and smash products with equivariant suspension spectra. 3.3. The slice tower G m With this definition it is possible to construct functors Pm+1 and PG with the same formal properties as in the classical case. Thus we get a tower

m+1 m m−1 ··· / P X / PG X / P X / ··· (2) G O O G O G m+1 G m G m−1 Pm+1 X Pm X Pm−1 X in which the homotopy colimit is X and the homotopy limit is contractible. G m We call this the slice tower. Pm X is the nth slice and the decreasing sequence of subgroups of π∗X is the slice filtration. We also get slice filtrations of the RO(G)-graded G H homotopy π⋆ X and the homotopy groups of fixed point sets π∗X . From now on we m m will drop the symbol G from the functors P , Pm and Pm . There is an important difference between this tower and the classical one. In the classical case the map X → P mX does not change homotopy groups in dimensions ≤ m. This is not true in this equivariant case.

38 A solution to the Arf-Kervaire invariant problem

m In the classical case, Pm X is an Eilenberg-Mac Lane spectrum whose nth homotopy m group is that of X. In our case, π∗Pm X need not be concentrated in dimension m. This means the slice filtration leads to a (possibly noncollapsing) slice spectral sequence converging to π∗X and its variants. One variant has the form s,t G t G E2 = πt−sPt X =⇒ πt−sX. G G Recall that π∗ X is by definition π∗X , the homotopy of the fixed point set. This is the spectral sequence we will use to study MU (G) and its relatives. 3.4. The Slice Theorem A large portion of our paper is devoted to proving that the slice spectral sequence has the desired properties. (G) 2m Slice Theorem. In the slice tower for MU , every odd slice is contractible and P2m decomposes as W2m ∧ HZ(2), where W2m is a certain wedge of 2m-dimensional slice cells (to be named later) and HZ is the 2-local integer Eilenberg-Mac Lane spectrum with c (2) c trivial G action. W2m never has any free summands. We will encounterc other spectra with similar properties, so we give them a name. Definition 3.2. A 2-local spectrum X is pure (isotropic) if each of its odd slices is contractible and each even slice has the form W ∧ HZ(2) where W is a wedge of pure (isotropic) cells (see Definition 3.1). c c In order to specify W2m, we need the following definition. Definition 3.3. Supposec X is a (2-local) G-spectrum such that its underlying homotopy u u group πk X is a free abelian group (a free Z(2)-module). A refinement of πk X is an equivariant map c : W → X in which W is a wedge of slice cells of dimensionc k whose underlying spheres represent a basis of πuX. ck u Recall that in π∗ (MUR), any monomial in the polynomial generators in dimension 2m is represented by an equivariant map from Smρ2 . u (C8) π∗ MU is a polynomial algebra with 4 generators in every positive even dimension. j We will denote the generators in dimension 2i by γ ri for 0 ≤ j ≤ 3. The action of 4 γ ∈ G = C8 is given by 4 i γ ri =(−1) ri.

u (C8) We will explain how π∗ MU can be reﬁned. u (C8) j π2 MU has 4 generators γ r1 that are permuted up to sign by G. It is reﬁned by an equivariant map (C8) W2 = S(ρ2) → MU . c b 39 HILL, HOPKINS and RAVENEL

2 Recall that the underlying spectrum of S(ρ2) is a wedge of 4 copies of S . In πuMU (C8) there are 14 monomials that fall into 4 orbits under the action of G, each 4 b corresponding to a map from a slice cell.

2 2 2 2 3 2 S(2ρ2) ←→ r1, γr1, γ r1, γ r1 2 2 3 3 Sb(2ρ2) ←→ r1 · γr1, γr1 · γ r1, γ r1 · γ r1, γ r1 · r1 2 3 Sb(2ρ2) ←→ r2, γr2, γ r2, γ r2 2 3 bS(ρ4) ←→ r1 · γ r1, γr1 · γ r1 b 4 4 u (C8) Recall that S(ρ4) is underlain by S ∨ S . It follows that π4 MU is reﬁned by an equivariant map from b

W4 = S(2ρ2) ∨ S(2ρ2) ∨ S(ρ4) ∨ S(2ρ2). c b b b b u (G) 3.5. Refining π∗ MU (G) More generally, we can proceed as follows. If we regard MU as a C2-spectrum, then u (G) kρ2 π∗ MU is refined by a map from a wedge of S s. More specifically, for each i> 0 we have a map

0 miρ2 (G) S [ri]= S → MU m_≥0

C2 (G) representing the powers of a certain generator ri ∈ π2i MU deﬁned in [HHR, §5.4.2]. Using the multiplication on MU (G), we get a map

0 0 (G) S [r1, r2,... ]= S [ri] → MU . i>^0

mρ2 This spectrum is a wedge of S s representing each monomial in the ris. g Now we apply the internal norm functor N2 to this map and get a map

g 0 (G) A = N2 S [r1, r2,... ] → MU (3) which reﬁnes the underlying homotopy of the target as a G-spectrum. A is a wedge of even dimensional slice cells, and the W2m in the Slice Theorem is the wedge of 2m-dimensional slice cells in A. c

3.6. The deﬁnition of ΩO

(G) Our G-spectrum ΩO (where G = C8) is obtained from the E∞-ring spectrum MU G by inverting a certain element D ∈ π19ρG speciﬁed in (6) below. The choices of G and D are the simplest ones leading to a homotopy ﬁxed point set with the detection property, (G) as shown in [HHR, §11]. The slice tower for ΩO has similar properties to that of MU .

40 A solution to the Arf-Kervaire invariant problem

4. The Gap Theorem 4.1. Derivation from the Slice Theorem G Assuming the Slice Theorem, the Gap Theorem follows from the fact that π−2 vanishes for each isotropic slice, i.e., for each one of the form

S(kρH ) ∧ HZ(2) for nontrivial H. Proving this amountsb to computing the homology of a certain chain complex. Details can be found in [HHR, §3.2]. These calculations do not require localization at 2, so we use HZ instead of HZ(2) thoughout this section. G kρG In order to give a feel for these calculations, a picture of π∗ S ∧ HZ for G = C8 and various integers k is shown in Figure 1. In order to derive the Gap Theorem from the Slice Theorem we need to ﬁnd the groups

G H mρh π∗ W (mρh) ∧ HZ = π∗ S ∧ HZ. We need this for all integers m because eventually we will obtain the spectrum ΩO by G (C8) inverting a certain element in π19ρ8 MU . Here is what we will learn. Vanishing Theorem.

H mρh • For m ≥ 0, πk S ∧ HZ = 0 unless m ≤ k ≤ mh. H mρh • For m < 0 and h > 1, πk S ∧ HZ = 0 unless mh ≤ k ≤ m − 2. The upper bound can be improved to m − 3 except in the case (h,m)=(2, −2) in which case H −2ρ2 we have π−4S ∧ HZ = Z(2).

H mρh Gap Corollary. For h> 1 and all integers m, πk S ∧ HZ = 0 for −4

C8 This means a similar statement must hold for π∗ ΩO = π∗Ω, which gives the Gap Theorem. Assuming the Slice Theorem, the Gap Theorem (the statement that π−2Ω = 0) follows immediately from the Gap Corollary above, as does the following.

G (G) Vanishing Line Corollary. The slice spectral sequence for π∗ MU is confined to the first quadrant and s,t E2 = 0 for s> (g − 1)(t − s), G (G) where g = |G|. If we invert any element in πkπG MU , the spectral sequence is confined to the first and third quadrants, and in the latter s,t E2 = 0 for s< min(0, (g − 1)(t − s + 4) + 4 − g). The proofs of the Vanishing Theorem and Gap Corollary are surprisingly easy. 4.2. An easy calculation

mρg We begin by constructing an equivariant cellular chain complex C(mρg)∗ for S , where m ≥ 0. In it the cells are permuted by the action of G, and the action of any

41 HILL, HOPKINS and RAVENEL

28 ◦ ◦ ◦ ⋄ ◦ ⋄ ◦ ⋄ ◦ ⋄ ⋄ ⋄ 14 ⋄ ◦ ⋄ ◦ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ k = −2 ⋄ ⋄ k = −4 ⋄ ⋄ ⋄ ⋄ k = 0 k = 2 k = 4 ⋄ ⋄ ⋄ ◦ ⋄ ◦ ⋄ ⋄ −14 ◦ ⋄ ◦ ⋄ ◦ ⋄ ◦ ◦ ◦ ◦ −32 −16 0 16 32

Figure 1. The homotopy of the slices Skρ8 ∧ HZ for small k. The horizontal and vertical axes are t − s and s in the slice spectral sequence. The numbers along the (t − s)-axis are even values of k. The groups G kρ8 π∗ S ∧ HZ lie along diagonals where t = 8k. Squares, bullets, circles and diamonds stand for Z, Z/2, Z/4 and Z/8 respectively. Lines of slopes 1, 3 and 7 separate regions with various types of torsion for even k. The gap can be seen in the center of this chart. The vanishing regions seen G (G) here also occur in the slice spectral sequence for π∗ MU .

subgroup H on the H-invariant cells is trivial. It is a complex of Z[G]-modules and is uniquely determined by ﬁxed point data of Smρg . For H ⊂ G we have

(Smρg )H = Smg/h.

This means there is a G-CW-complex with one cell in dimension m, two cells in each dimension from m +1to2m, four cells in each dimension from 2m +1to4m, and so on.

42 A solution to the Arf-Kervaire invariant problem

In other words, 0 unless m ≤ k ≤ gm C(mρg)k =  Z for k = m  Z[G/H] for mg/2h

Then we have G mρg π∗ (S ∧ HZ)= H∗(HomZ[G](Z,C(mρg))). These groups are nontrivial only for m ≤ k ≤ gm, which gives the Vanishing Theorem for m ≥ 0. We will look at the bottom three groups in the complex HomZ[G](Z,C(mρg)∗). Since C(mρg)k is a cyclic Z[G]-module, the Hom group is always Z. For m> 1 our chain complex C(mρg) has the form

C(mρg)m C(mρg)m+1 C(mρg)m+2

ǫ 1−γ 1+γ 0 o Z o Z[C2] o Z[C2] o ...

Applying HomZ[G](Z, ·) to this gives (in dimensions ≤ 2m for m> 4)

Zo 2 Z o 0 Z o 2 Z o 0 Z o ... m m + 1 m + 2 m + 3 m + 4 It follows that for m ≤ k < 2m, Z/2 k ≡ m mod 2 πG(Smρg ∧ HZ) = k 0 otherwise.

−mρg For negative multiples of ρg, S (with m> 0) is the equivariant Spanier-Whitehead dual of Smρg . This means that

G −mρg ∗ π∗ S ∧ HZ = H (HomZ[G](C(mρg), Z)).

Applying the functor HomZ[G](·, Z) to our chain complex C(mρg)

ǫ 1−γ 1+γ 1−γ Zo Z[C2] o Z[C2] o Z[C2 or C4] o ... m m + 1 m + 2 m + 3 gives a cochain complex beginning with

Z 1 / Z 0 / Z 2 / Z 0 / Z / ... −m −m − 1 −m − 2 −m − 3 −m − 4

43 HILL, HOPKINS and RAVENEL

Here is a diagram showing both functors in the case m ≥ 4.

m m + 1 m + 2 m + 3 m + 4 Zo Z o Z o Z o Z o ... 2 0 KS 2 0 Z HomZ[G]( , ·)

ǫ 1−γ 1+γ 1−γ 1−γ Zo Z[C2] o Z[C2] o Z[C2] o Z[C2] o ...

Z HomZ[G](·, ) Z 1 / Z 0 / Z 2 / Z 0 / Z / ... −m −m − 1 −m − 2 −m − 3 −m − 4

Note the diﬀerence in the behavior of the map ǫ : Z[C2] → Z under the two functors HomZ[G](Z, ·) and HomZ[G](·, Z). They convert it to maps of degrees 2 and 1 respectively. This diﬀerence is responsible for the gap.

5. The Periodicity Theorem We now outline the proof of the Periodicity Theorem (assuming the Slice Theorem), which is treated in [HHR, §9]. We establish some diﬀerentials in the slice spectral sequence and show that certain G (G) elements become permanent cycles after inverting a certain D ∈ π19ρG MU (deﬁned below in (6)), where G = C8. This leads to an equivariant self map

256 Σ ΩO → ΩO.

It is an ordinary homotopy equivalence, and we will see that this implies formally that it induces an equivalence on homotopy ﬁxed point sets.

5.1. Geometric fixed points The key tool for studying differentials in the slice spectral sequence is the geometric fixed point spectrum, denoted by ΦGX for a G-spectrum X. A detailed account and references can be found in [HHR, §2.5]. It has much nicer properties than the usual fixed point spectrum XG, which is awkward for two reasons: • it fails to commute with smash products and • it fails to commute with infinite suspensions. The geometric fixed set ΦGX is a convenient substitute that avoids these difficulties. In order to define it we need the isotropy separation sequence, which is the cofiber sequence

0 EP+ → S → E˜P. (4)

44 A solution to the Arf-Kervaire invariant problem where EP is a G-space such that for a subgroup H of G, empty for H = G EPH = contractible otherwise. Such a space can be constructed for any finite group G and it is unique up to equivariant homotopy equivalence. For cyclic 2-group G, the space EC2 with G acting via it projection onto C2 has the required properties. The symbol P here stands for the family of proper subgroups of G. A similar space EF can be constructued for any family of subgroups F closed under inclusion and conjugation. Definition 5.1. For a G-spectrum X the geometric fixed point set is ΦGX =(E˜P∧ X)G. This functor has the following properties: Theorem 5.2. (i) For G-spectra X and Y , ΦG(X ∧ Y ) = ΦGX ∧ ΦGY . (ii) For a G-space X, ΦGΣ∞X = Σ∞(XG). (iii) A map f : X → Y is a G-equivalence iff ΦH f is an ordinary equivalence for each subgroup H ⊂ G. From the suspension property we can deduce that for any finite cyclic 2-group G, ΦGMU (G) = MO, the unoriented cobordism spectrum. Its homotopy type has been well understood since Thom’s work [Tho54] in the 50s. Geometric Fixed Point Theorem. Let G be a finite cyclic 2-group and let σ denote ˜ −1 its sign representation. Then for any G-spectrum X, π⋆EC2 ∧ X = aσ π⋆X, where 0 σ aσ ∈ π−σX is induced by the inclusion of the fixed point set S → S . In [HHR, §5.4.2] we define specific polynomial generators rG ∈ πC2 MU (G) (5) i iρ2 that are convenient for our purposes. We denote the underlying homotopy classes by ri u (G) that lie in π2iMU . k Recall that π∗MO = Z/2[yi : i > 0,i 6= 2 − 1] where |yi| = i. It is not hard to show that u (C8) 2 3 π∗ MU = Z(2)[ri,γ(ri),γ (ri),γ (ri): i> 0] 4 i (4) where |ri| = 2i, γ is a generator of G and γ (ri)=(−1) ri. In πiρ8 MU we have the element 2 3 Nri = riγ(ri)γ (ri)γ (ri).

G 8 iρ8 (C8) i Applying the functor Φ to the map N2 ri : S → MU gives a map S → MO.

45 HILL, HOPKINS and RAVENEL

Lemma 5.3. The generators ri and yi satisfy k G g 0 for i = 2 − 1 Φ N2 ri = yi otherwise. 5.2. Some slice diﬀerentials We know that the slice spectral sequence for MU (G) has a vanishing line of slope g −1. We will describe the subring of elements lying on it. (G) Let fi ∈ πi(MU ) be the composite

aiρg Nri Si / Siρg / MU (G),

where aiρg is the inclusion of the ﬁxed point set. The following facts about fi are easy to prove. (g−1)i,gi • It appears in the slice spectral sequence in E2 , which is on the vanishing line. • The subring of elements on the vanishing line is the polynomial algebra on the fi. • Under the map

(G) G (G) π∗MU → π∗Φ MU = π∗MO we have 0 for i = 2k − 1 fi 7→ yi otherwise • Any diﬀerential landing on the vanishing line must have a target in the ideal given by 2kσ (f1,f3,f7,... ). A similar statement can be made after smashing with S . |V | V For an oriented representation V there is a map uV : S → Σ HZ(2), which lies in − 2k 1 πV −|V |(HZ(2)). It satisﬁes uV +W = uV uW , so u2kσ = u2σ .

k Slice Diﬀerentials Theorem. In the slice spectral sequence for Σ2 σMU (G) for k > 0, k we have dr(u2kσ) = 0 for r< 1+ g(2 − 1), and

2k d1+g(2k−1)(u2kσ)= aσ f2k−1.

Sketch of proof: Inverting aσ in the slice spectral sequence will make it converge to π∗(MO). This means each power of u2σ has to support a nontrivial diﬀerential. The only way this can happen is as indicated in the theorem.

Typically one proves theorems about diﬀerentials in such spectral sequences by means of some sort of extended power construction. In our case, all of the necessary geometry u (G) is encoded in the relation between π∗ MU and π∗MO!

46 A solution to the Arf-Kervaire invariant problem

5.3. Some RO(G)-graded calculations For a cyclic 2-group G let dG g g/2−1 k = N2 r2k−1 = r2k−1γ(r2k−1) ...γ (r2k−1) (G) k ∈ π(2 −1)ρg MU We want to invert this element and study the resulting slice spectral sequence. As explained previously, for G = C8 it is confined to the first and third quadrants with vanishing lines of slopes 0 and 7. 2k The differential dr on u2σ described in the theorem is the last one possible since its 2k+1 target, aσ f2k+1−1, lies on the vanishing line. If we can show that this target is killed dG 2k by an earlier differential after inverting k , then u2σ will be a permanent cycle. We have G 2k+1−1 g g k+1 d k+1 k f2 −1 k = (aρg N2 r2 −1)(N2 r2 −1) 2k g 2k−1 g k+1 k = aρg N2 r2 −1(aρg N2 r2 −1) 2k G d k = aρg k+1f2 −1 2k dG 2k = aV k+1aσ f2k−1 where V = ρg − σ 2k dG = aV p k+1d1+8(2k−1)(u2kσ). −1 dG (G) Corollary. In the RO(G)-graded slice spectral sequence for k MU , the class 2k given by u2k+1σ = u2σ is a permanent cycle. 5.4. An even trickier RO(G)-graded calculation

The Corollary shows that inverting a certain element makes a power of u2σ a permanent cycle. We need to invert something to make a power of u2ρ8 a permanent cycle. We will get this by using the norm property of u. It says that if V is an oriented representation of a subgroup H ⊂ G with V H = 0 and V ′ is the induced representation g g ′′ ′ of V , then the norm functor Nh from H-spectra to G-spectra satisﬁes Nh (uV )uV = uV , where V ′′ is the induced representation of the trivial representation of degree |V |. From this we can deduce that 8 8 u2ρ8 = u8σ8 N4 (u4σ4 )N2 (u2σ2 ), where σg denotes the sign representation on Cg. By the Corollary we can make a power of each factor a permanent cycle by inverting dC2m some km for 1 ≤ m ≤ 3. If we make km too small we will lose the detection property, that is we will get a spectrum that does not detect the θj. It turns out that km must be m chosen so that 8|2 km. dC2 • Inverting 4 makes u32σ2 a permanent cycle.

47 HILL, HOPKINS and RAVENEL

dC4 • Inverting 2 makes u8σ4 a permanent cycle. dC8 • Inverting 1 makes u4σ8 a permanent cycle. 16 • Inverting the product D of the norms of all three makes u32ρ8 = u2ρ8 a permanent cycle. Let C C C d 8 8 d 4 8 d 2 8 C8 C4 C2 (C8) D = 1 N4 ( 2 )N2 ( 4 )= N2 (r1 r3 r15 ) ∈ π19ρ8 MU . (6) Then we deﬁne

−1 (C8) hC8 ΩO = D MU and Ω = ΩO . (7) Since the inverted element is represented by a map from Smρ8 , the slice spectral C8 sequence for π∗Ω= π∗ ΩO has the usual properties: • It is concentrated in the ﬁrst and third quadrants and conﬁned by vanishing lines of slopes 0 and 7. • It has the gap property, i.e., no homotopy between dimensions −4 and 0.

5.5. The proof of the Periodicity Theorem 2 16 (8) dC8 16,0 (8) Preperiodicity Theorem. Let ∆1 = u2ρ8 1 ∈ E2 (ΩO). Then ∆1 is a permanent cycle.

16 32 (8) dC8 dC8 To prove this, note that ∆1 = u32ρ8 1 . Both u32ρ8 and 1 are permanent 16 (8) cycles, so ∆1 is also one. 256 Hence we have an equivariant map Π : Σ ΩO → ΩO where

256−32ρ8 0 • u32ρ8 : S → ΩO induces to the unit map from S on the underlying ring spectrum and (8) • ∆1 is invertible because it is a factor of D.

The above imply that the underlying map i0Π of ordinary spectra is a homotopy equivalence. It is known that any such map induces an equivalence of homotopy ﬁxed point sets, so

hC8 256 hC8 Π hC8. O Σ ΩO ≃ / Ω Unfortunately the slice spectral sequence tells us nothing about this homotopy fixed point set. We know it detects all of the θj, but there is no direct way of showing that it has the gap property. Fortunately we have a theorem (the subject of the next section) stating that in this case the homotopy fixed set is equivalent to the actual fixed point set Ω. The slice spectral sequence tells us that the latter has the gap property. Thus we have proved Periodicity Theorem. For Ω as in (7), Σ256Ω is equivalent to Ω.

48 A solution to the Arf-Kervaire invariant problem

6. The Homotopy Fixed Point Theorem In order to finish the proof of the main theorem, we need to show that the actual fixed −1 (C ) point set of ΩO = D MU 8 is equivalent to its homotopy fixed point set. A detailed account can be found in [HHR, §10]. The slice spectral sequence computes the homotopy of the former while the Hopkins-Miller spectral sequence, also known as the homotopy fixed point spectral sequence (which is known to detect θj), computes that of the latter. Here is a general approach to showing that actual and homotopy fixed points are equivalent for a G-spectrum X. 0 We have an equivariant map EG+ → S . Mapping both into X gives a map of G G-spectra ϕ : X+ → F (EG+,X+). Passing to fixed points would give a map from X to XhG, which we would like to be an equivalence. We will prove the stronger statement that ϕ is a G-equivalence. The case of interest is X = ΩO and G = C8. We will argue by induction on the order of the subgroups H of G, the statement being obvious for the trivial group. We will smash ϕ with the cofiber sequence 0 EG+ → S → EG.˜ This gives us the following diagram in which both rows are cofiber sequences.

EG+ ∧ ΩO / ΩO / EG˜ ∧ ΩO

′ ′′ ϕ ϕ ϕ EG+ ∧ F (EG+, ΩO) / F (EG+, ΩO) / EG˜ ∧ F (EG+, ΩO) ′ The map ϕ is an equivalence because ΩO is nonequivariantly equivalent to F (EG+, ΩO), and EG+ is built up entirely of free G-cells. Thus it suffices to show that ϕ′′ is an equivalence, which we will do by showing that both its source and target are contractible. Both have the form EG˜ ∧ X where X is a module spectrum over ΩO, so it suffices to show that EG˜ ∧ ΩO is contractible. We will show that it is H-equivariantly contractible by induction on the order of the subgroups H of G. Over the trivial group EG˜ itself is contractible. Let H be a subgroup, ′ ′ H ⊂ H the subgroup of index 2 and H2 = H/H . We will smash our spectrum with the isotropy separation sequence of (4), which in our case is 0 EH2+ → S → EH˜ 2. ′ Then EH˜ 2 ∧ EG˜ ∧ ΩO is contractible over H , so it suffices to show that its H-fixed point set is contractible. It is H H H Φ (EG˜ ∧ ΩO) = Φ (EG˜ ) ∧ Φ (ΩO), H H and Φ (ΩO) is contractible because Φ (D) = 0. Thus it remains to show the ′ H-contractibility of EH2+ ∧ EG˜ ∧ ΩO. But this is equivalent to the H -contractibility of EG˜ ∧ ΩO, which we have by induction.

49 HILL, HOPKINS and RAVENEL

Sidebar 3 Formal group laws I

Formal groups laws are are key tool in complex cobordism theory, which is part of the infrastructure of our proof. The deﬁnitive reference is Hazewinkel’s book [Haz78]. A much briefer account can be found in [Rav86, Appendix 2]. A formal group law F over a ring R is a power series F (x,y) ∈ R[[x,y]] satisfying three conditions, namely (i) Commutativity F (y,x)= F (x,y) (ii) Identity F (x, 0) = F (0,x)= x (iii) Associativity F (F (x,y),z)= F (x,F (y,z)). Elementary examples include the additive and multiplicative formal group laws, x + y and 1+ x + y + xy respectively. When R is torsion free, F is determined by a power series over R ⊗ Q in one variable called the logarithm of F satisfying

logF (F (x,y)) = logF (x) + logF (y). n n+1 For our two elementary examples it is x and n≥0(−1) x /(n + 1). P There is a universal example due to Lazard [Laz55] of a formal group law G over a ring L with the property that for any formal group law F over any ring R, there is a unique ring homomorphism θ : L → R sending G to F . It is given by i j G(x,y)= ai,j x y and L = Z[ai,j ]/I Xi,j

where I is the ideal generated by the relations among the coeﬃcients ai,j forced by conditions (i)-(iii) above. To describe L explicitly, it is useful to give it a grading such that G(x,y) is homogeneous of degree 2 if |x| = |y| = 2. This means that |ai,j | = 2(1 − i − j). Lazard then shows that

L = Z[xi : i> 0] where |xi| = −2i.

7. The Detection Theorem The Detection Theorem is the subject of [HHR, §11].

7.1. θj in the Adams-Novikov spectral sequence

Browder’s Theorem says that θj is detected in the classical Adams spectral sequence by 2 2,2j+1 hj ∈ ExtA (Z/2, Z/2), where A denotes the mod 2 Steenrod algebra. This element is known to be the only one in its bidegree.

50 A solution to the Arf-Kervaire invariant problem

Sidebar 4 Formal group laws II

There is a formal group law F defined over the complex cobordism ring MU ∗ defined as follows. We know that MU ∗(CP ∞)= MU ∗[[x]] where x ∈ MU 2CP ∞ is the first Chern class of the canonical complex line bundle. Similarly, MU ∗(CP ∞ × CP ∞)= MU ∗[[x ⊗ 1, 1 ⊗ x]]. There is a map CP ∞ × CP ∞ → CP ∞ related to the tensor product of line bundles. It sends x to a power series F (x ⊗ 1, 1 ⊗ x), which can easily be shown to be a formal group law. Quillen [Qui69] showed that the Lazard classifing map L → MU ∗ is an isomorphism.

Cartier [Car67] showed that when R is a torsion free Z(p)-algebra, F is canonically isomorphic to a formal group law F ′ which is p-typical, meaning that its logarithm has the form pn logF ′ (x)= x + ℓnx ∈ R ⊗ Q[[x]]. n>X0 The analog of the Lazard ring for such groups is much smaller than L, having polyno- n mial generators vn only in each dimension of the form 2(1 − p ).

The topological analog of Cartier’s theorem is the splitting of MU(p) into a wedge of suspensions of spectra known as BP , originally constructed by Brown and Peterson [BP66a].

It is more convenient for us to work with the Adams-Novikov spectral sequence, which maps to the Adams spectral sequence. It has a family of elements in filtration 2, namely 2,6i−2j βi/j ∈ ExtMU∗(MU) (MU∗,MU∗) for certain values of of i and j. When j = 1, it is customary to omit it from the notation. The definition of these elements can be found in [Rav86, Chapter 5]. Here are the first few of these in the relevant bidegrees. 2,16 θ3 : β4/4 and β3 ∈ ExtMU∗(MU) (MU∗,MU∗) , 2,32 θ4 : β8/8 and β6/2 ∈ ExtMU∗(MU) (MU∗,MU∗) , 2,64 θ5 : β16/16, β12/4 and β11 ∈ ExtMU∗(MU) (MU∗,MU∗) , 2 and so on. In the bidegree of θj, only β2j−1/2j−1 has a nontrivial image (namely hj ) in the Adams spectral sequence. There is an additional element in this bidegree, namely α1α2j −1. Thus 2,2j+1 ExtMU∗(MU) (MU∗,MU∗)

51 HILL, HOPKINS and RAVENEL

2 is an elementary abelian 2-group of rank [(j + 3)/2]. An element in it maps to hj in the corresponding classical Adams Ext group iff the coefficient of β2j−1/2j−1 (with respect to the evident basis) is nontrivial. We need to show that any such element has nontrivial image the Adams-Novikov spectral sequence for Ω. 2,2j+1 Detection Theorem. Let j ≥ 6 and let x ∈ ExtMU∗(MU) (MU∗,MU∗) be any element j+1 2,2 2 2,2j+1 whose image in ExtA (Z/2, Z/2) is hj . Then the image of x in H (C8; π∗ΩO) is nonzero. We will prove this by showing the same is true after we map the latter to a simpler Ext group defined in terms of another algebraic tool, the theory of formal A-modules, where A is the ring of integers in a suitable field. In that setting the group in the relevant bidegree 4 is (Z/8) . We will show that β2j−1/2j−1 maps to an element of order 2, while the other basis elements map to 0. 7.2. Formal A-modules Recall the a formal group law (see Sidebar 4) over a ring R is a power series i j F (x,y)= x + y + ai,j x y ∈ R[[x,y]] i,j>X0 with certain properties. For positive integers m one has power series [m](x) ∈ R[[x]] defined recursively by [1](x)= x and [m](x)= F (x, [m − 1](x)). These satisfy [m + n](x)= F ([m](x), [n](x)) and [m]([n](x))=[mn](x). With these properties we can define [m](x) uniquely for all integers m, and we get a homomorphism τ : Z → End(F ) m 7→ [m](x) from the integers to the endomorphism ring of F . If the ground ring R is an algebra over the p-local integers Z(p) or the p-adic integers Zp, then we can make sense of [m](x) for m in Z(p) or Zp. Now suppose R is an algebra over a larger ring A, such as the ring of localized integers in a number field or a finite extension of the p-adic numbers. We say that the formal group law F is a formal A-module if the homomorphism τ extends to A in such a way that [a](x) ≡ ax mod (x2) for a ∈ A. The theory of formal A-modules is well developed. Lubin-Tate [LT65] used it to do local class field theory.

52 A solution to the Arf-Kervaire invariant problem

The example of interest to us is A = Z2[ζ8], where ζ8 is a primitive 8th root of unity. It is the ring of integers in a cyclotomic extension of the 2-adic numbers Q2. The maximal 4 ideal of A is generated by π = ζ8 − 1, and π is a unit multiple of 2. There is a formal ±1 A-module F over R∗ = A[w ] (with |w| = 2) satisfying

logF (F (x,y)) = logF (x) + logF (y) where

n n w2 −1x2 log (x)= . (8) F πn nX≥0

We need to consider the cohomology of C8 with coeﬃcients in R∗, where the action ∗ of the group is trivial on A and γw = ζw for a group generator γ. H (C8; R∗) is the cohomology of the cochain complex

γ−1 Trace γ−1 R∗ / R∗ / R∗ / ... where Trace is multiplication by 1 + γ + ··· + γ7. s The cohomology groups H (C8; R∗) for s> 0 are periodic in s with period 2. The part of it that matters to us is

2 8m 8m 2 3 H (C8; R16m)= w A/(8) = w Z/(8) 1, π, π , π .

7.3. The relation between MU (C8) and formal A-modules −1 (C ) What does this have to do with our spectrum ΩO = D MU 8 ? Recall that D is dC8 8 dC4 8 dC2 given by 1 N4 ( 2 )N2 ( 4 ). We saw earlier that inverting a product of this sort is needed to get the Periodicity Theorem, but we did not explain the choice of subscripts of d. They are the smallest ones that satisfy the second part of the following.

Lemma 7.1. The classifying homomorphism λ : π∗MU → R∗ for F factors through (C8) π∗MU in such a way that

(4) u (C8) • the homomorphism λ : π∗ MU → R∗ is equivariant, where C8 acts on (C8) π∗MU as before, it acts trivially on A and γw = ζ8w for a generator γ of C8. (C8) • The element D ∈ π∗MU that we invert to get ΩO goes to a unit in R∗.

We will sketch the proof of this later.

53 HILL, HOPKINS and RAVENEL

7.4. Deriving the Detection Theorem from Lemma 7.1 Lemma 7.1 implies that there is a diagram

S0 / MU (C8)

s,t s,t (C8) ExtMU∗(MU) (MU∗,MU∗) / ExtMU∗(MU) MU∗,MU∗(MU )

s u (C8) H (C8; πt MU ) ∗ WWWWW λ(4) WWWWW WWWW+ s s −1 (C8) s H (C8; πtΩO) H (C8; πtD MU ) / H (C8; Rt) where the unit map at the top induces the homomorphism below it. We prove the Detection Theorem by showing that the composite map

j+1 2,2 2 j+1 ExtMU∗(MU) (MU∗,MU∗) / H (C8; R2 )

2j Z/2 α1α2j −1, β2j−1/2j−1 , β3·2j−3/2j−3 , ... w A/(8) sends β2j−1/2j−1 to a nontrivial element of order 2, and each of the other basis elements to 0. This is a calculation requiring tools from BP -theory which is spelled out in [HHR, §11.4].

7.5. The proof of Lemma 7.1 To prove the ﬁrst part, consider the following diagram for an arbitrary ring K.

u (C4) π∗ MU

(2) π∗MU

MU∗(MU) 5 iR ηL ll RRRηR llll RRR l λ(2) π∗MU π∗MU SS k SSSS kkkk λ1 SSS kkk λ2 S) K ku k

The maps λ1 and λ2 classify two formal group laws F1 and F2 over K. The Hopf alge- broid MU∗(MU) represents strict isomorphisms between formal group laws. Hence the (2) existence of λ is equivalent to that of a strict isomorphism between F1 and F2.

54 A solution to the Arf-Kervaire invariant problem

Similarly consider the following diagram.

u (C8) π∗ MU

(4) π∗MU η 3 7 g kV η 1 hhhhoo OOOVVVV 4 hhhh oo OO VVVV hhhh oooη2 η3 OO VVVV hh (4) V π∗MU π∗MU λ π∗MU π∗MU WWW P λ2 λ3 gg WWWW PP nn gggg WWWW PPP nnn gggg λ1 WWWWPP nngggg λ4 WW+' K sgwngg

(4) The four maps ηj are induced by the four unit maps from MU to MU , the spectrum underlying MU (C8). The existence of λ(4) is equivalent to that of strict isomorphisms between the formal group laws Fj classiﬁed by the λj. Now suppose that K = R∗ and each Fj is the formal group law given by (8). Then (4) we have the required isomorphisms, so λ exists. The inclusions ηj are related to the (4) C8-action on π∗MU by

γηj = ηj+1 for 1 ≤ j ≤ 3, i and γη4 diﬀers from η1 by (−1) in dimension 2i. The λj can be chosen so that

γλj = λj+1 for 1 ≤ j ≤ 3.

(4) It follows that λ is equivariant with respect to the C8-actions on its source and target. This proves the ﬁrst part of Lemma 7.1. For the second part, recall that

8 C2 C4 C8 D = N2 (r15 r3 r1 ). The norm sends products to products, and N(x) is a product of conjugates of x under the action of C8. Hence its image in R∗ is a unit multiple of that of a power of x, so it C2 C4 C8 suﬃces to show that each of the three elements r15 , r3 and r1 maps to a unit in R∗. We refer the reader to [HHR, §11.5] for the details.

8. The Slice and Reduction theorems Recall that a pivotal step in our proof is the Slice Theorem, which identiﬁes the layers in the slice tower for MUR and its relatives. In [HHR, §6] it is derived from the Reduction Theorem, which we now state. For each cyclic 2-group G = C2n+1 there is a certain equivariant (noncommutative) ring spectrum A, deﬁned in (3), which is a certain wedge of slice cells. It maps to MU (G) in such a way that the underlying wedge of spheres hits all of the underlying homotopy of MU (G). We can map it to S0 (with trivial group action) by sending all of the 0-connected summands to a point. Thus both MU (G) and S0 are A-modules.

55 HILL, HOPKINS and RAVENEL

(G) (G) 0 Reduction Theorem. Let R(∞) = MU ∧A S . The map (G) fG : R(∞) → HZ(2) given by the 0th slice is an equivariant equivalence.

(G) Recall that we have localized MU at the prime 2. We will denote R(∞) and fG simply by R(∞) and f when the value of G is clear from the context. The nonequivariant version of this is obvious. It says that if we kill of all of the (G) positive dimensional homotopy of the spectrum underlying MU , we get HZ(2), its zeroth Postnikov section. The case G = C2 was proved by Hu-Kriz in [HK01]. The derivation of the Slice Theorem from the Reduction Theorem is essentially formal. The proof of the Reduction Theorem is the hardest calculation in our paper and is treated in [HHR, §7].

8.1. Deriving the Slice Theorem from the Reduction Theorem

It is possible to define an A-module structure on Md, the wedge of all (d−1)-connected summands of A. Then M2d−1 = M2d and W2d (defined in 3.5) is the cofiber of the inclusion M → M , which is also an A-module. 2d+2 2d c Let (G) d (G) Kd = MU ∧A Md and P˜ = MU /Kd−1 2d 2d−1 Then it is not difficult to show that the fiber of the inclusion P˜ → P˜ is R(∞)∧W2d. The Reduction Theorem implies that this is HZ ∧ W . It also enables us to identify (2) 2d c the diagram c ··· / P˜2d+1 / P˜2d / P˜2d−1 / ··· O O O

∗ R(∞) ∧ W2d ∗ c

∧W2d with the slice tower (2) for MU (G), i.e., toc show that P˜m is equivalent to P mMU (G). The Slice Theorem follows. There is a converse [HHR, Prop. 6.9] which says roughly that if a ring spectrum R (such as MU (G) ∧ MU (G)) with properties similar to those of MU (G) satisfies the Slice Theorem, then the Reduction Theorem follows. Since this converse is used in the proof of the Reduction Theorem itself, we will state it more precisely. u For a finite cyclic 2-group G, let R be a connective 2-local G-ring spectrum with π∗ R a polynomial algebra over Z(2) on positive dimensional generators. We suppose further u that π∗ R is refined by a map from a spectrum A of the form g 0 A = N2 S [x1, x2,... ]

56 A solution to the Arf-Kervaire invariant problem where x ∈ πC2 X with k > 0 for each i. Then we can deﬁne M , K and P˜d for R as i kiρ2 i d d we did for MU (G) above. In particular P˜0 would be the analog of R(∞) as above. Then similar arguments to those above give ˜0 Proposition 8.1. For R as above, if P is equivalent to HZ(2), then R is pure and isotropic. Here is our converse. Proposition 8.2. Let R be a connective 2-local G-ring spectrum as above, and suppose that it is pure (Deﬁnition 3.2). Then the tower P˜d is the slice tower for R. In ˜0 0 n o particular, P is P R = HZ(2). 8.2. The proof of the Reduction Theorem

We will use a double induction argument. Assuming that the map fG (which is defined only for nontrivial G) is H-equivariant for a subgroup H ⊆ G, we will show that (i) When H is a proper subgroup of G, it can be replaced by the next larger subgroup of G. (ii) G can be replaced by the next larger supergroup G′ of G. We have the statement for trivial H and any G since that is the nonequivariant case. Hence the Reduction Theorem will follow from the two inductive steps. The second inductive step follows immediately from the fact that the restriction of ′ R(∞)(G ) to G is R(∞)(G). This means that we need only show the first step in the case where H is the index two subgroup of G. For A as in (3) and an integer i ≥ 0 we define an A-module (G) (G) ′ ′ g 0 R(i) = MU ∧A Ai where Ai = N2 S [ri+1, ri+2,... ]. (9) Roughly speaking, R(i)(G) is obtained from MU (G) by killing the subring of the underlying homotopy generated by elements below dimension 2i. We will often denote it simply by R(i). Here are three consequences of the inductive hypothesis, proved in [HHR, Propositions 7.2 and 7.3]. Proposition 8.3. Let H ⊂ G be the index 2 subgroup. If the inductive hypothesis above holds in this case, then (i) The Slice Theorem holds for H when |H| > 1. ∗ (G) (ii) iH MU is a pure isotropic spectrum. ∗ (G) (iii) iH R(i) (see (9)) is pure and isotropic. ∗ (G) (H) Proof. (i) Since iH R(∞) = R(∞) , the hypothesis of this proposition is the same as the Reduction Theorem for H. It implies the Slice Theorem for H. (ii) MU (G) and MU (H) ∧ MU (H) are underlain by the same smash power of MU. ∗ (G) (H) Thus iH MU is a wedge of suspensions of MU , and this splitting can be shown to ∗ (G) (H) be equivariant. Hence iH MU is pure isotropic because MU is.

57 HILL, HOPKINS and RAVENEL

∗ (G) ∗ (G) (iii) iH R(i) has an equivariant splitting similar to that of iH MU in which each summand is a suspension of R(i)(H). The reduction theorem for H implies that R(i)(H) ∗ (G) is pure and isotropic by Proposition 8.1, so the same is true of iH R(i) .

Now we smash the map of the Reduction Theorem with the isotropy separation sequence of (4) and get a diagram

EP+ ∧ R(∞) / R(∞) / E˜P∧ R(∞) ′ ′′ f f f ˜ EP+ ∧ HZ(2) / HZ(2) / EP∧ HZ(2) We want to show that f is an equivariant equivalence. The inductive hypothesis implies that f ′ is one, so it suﬃces to show that f ′′ is one. We see from Theorem 5.2 (iii) this boils down to showing that the map G G ′′ G G π∗Φ f = π∗ f : π∗Φ R(∞) → π∗Φ HZ(2) (10) is an isomorphism. It is not hard to show that the source and target of (10) are abstractly isomorphic and of ﬁnite type [HHR, Prop. 7.5]. Proposition 8.4. G −2 G −1 G π∗Φ HZ(2) = Z/2[b] where b = u2σaσ ∈ π2Φ HZ(2) ⊂ aσ π⋆ HZ(2). and Z/2 for n ≥ 0 and even π ΦGR(∞)= n 0 otherwise.

G (G) Note that π∗Φ R(∞) lacks a multiplication because it is not obvious that R(∞) is a ring spectrum. We only know it is as a consequence of the Reduction Theorem. Proof. The first assertion follows from computations similar to those of 4.2. G (G) For the second we will compute π∗Φ R(i) by induction on i. Since R(0) = MU , ΦGR(0) = MO, so G k π∗Φ R(0) = π∗MO = Z/2[hj ,j 6= 2 − 1] G G where hj ∈ πjMO. We can identify the fiber of the map Φ R(i − 1) → Φ R(i) and we have a cofiber sequence

ΦGr ΣkΦGR(i − 1) i / ΦGR(i − 1) / ΦGR(i) where h for i 6= 2k − 1 ΦGr = i (11) i 0 for i = 2k − 1. G The indicated value of π∗Φ R(∞) follows.

58 A solution to the Arf-Kervaire invariant problem

Thus it suﬃces to show that the homomorphism of (10) is onto. This job is simpliﬁed by the following, which is [HHR, Lemma 7.7]. Let g 0 (G) 0 Bi = N2 S [ri] and Xi = MU ∧Bi S . (12) (G) Xi is the spectrum obtained from MU by killing the polynomial generators in dimension 2i of its underlying homotopy.

k−1 Lemma 8.5. If for every k ≥ 1, the class b2 is in the image of G (G) 0 G π k Φ MU ∧ − S → π k Φ HZ , 2 B2k 1 2 (2) then the map of (10) is onto. Proof. We have

R(∞)= X1 ∧MU (G) X2 ∧MU (G) ··· k−1 This implies that if b2 is in the image for each k ≥ 1, then every power of b is.

k Let ck = 2 − 1. To show that the hypothesis of Lemma 8.5 is met we need to study the element g G (G) N2 rck ∈ πckρG MU , k−1 which we will see to be related to an element mapping to b2 under the homomorphism G ˜ of Lemma 8.5. Its image in πckρG EP ∧ Xck is zero for two reasons: we are coning it oﬀ G to form Xck , and it is killed by the functor Φ . We will verify the hypothesis of Lemma 8.5 by chasing this element around the diagram

(G) (G) (G) EP+ ∧ MU / MU / E˜P∧ MU (13)

˜ EP+ ∧ Xck / Xck / EP∧ Xck

˜ EP+ ∧ HZ(2) / HZ(2) / EP∧ HZ(2) Here the rows are coﬁber sequences, but the columns are not. Since SiρG is obtained from Si by attaching induced cells, one can show [HHR, Remark 2.34] that the restriction map G ˜ G ˜ πiρG EP∧ X → πi EP∧ X g is an isomorphism for any G-spectrum X. The image of N2 rck in G ˜ (G) G ˜ (G) πckρG EP∧ MU ≈ πck EP∧ MU ≈ πck MO g is zero by (11). Thus in the top row of (13), N2 rck has a preimage G (G) yk ∈ πckρG EP+ ∧ MU .

59 HILL, HOPKINS and RAVENEL

Proposition 8.6. Any choice of yk as above has nontrivial image in G πckρG EP+ ∧ HZ(2). Proof of the Reduction Theorem modulo Proposition 8.6. We have already seen that the Reduction Theorem follows from the hypothesis of Lemma 8.5, that bck is in the image of G G the map induced by f. The image of yk in πckρG EP+ ∧Xck has trivial image in πckρG Xck , so it has a preimage G ˜ y˜k ∈ πckρG+1EP+ ∧ Xck . G ˜ The image ofy ˜k in πckρG+1EP+ ∧ HZ(2) is nontrivial by Proposition 8.6. Now consider the diagram

G ˜ ≈ G ˜ πckρG+1EP+ ∧ Xck / π2k EP+ ∧ Xck

G ˜ ≈ G ˜ πckρG+1EP+ ∧ HZ(2) / π2k EP+ ∧ HZ(2)

k where the horizontal maps are restrictions along the inclusion S2 → SckρG+1. The lower − 2k 1 right group is Z/2 generated by b . We have just shown thaty ˜k maps nontrivially to k−1 the lower left group, so b2 is in the image of the map induced by f.

Proposition 8.6 concerns the map

(G) EP+ ∧ MU → EP+ ∧ HZ(2). Each cell of both source and target is induced up from the index 2 subgroup H, so we can use the inductive hypothesis. We will use the factorization

(G) k (G) EP+ ∧ MU → EP+ ∧ R(2 − 2) → EP+ ∧ HZ(2),

k (G) G k (G) with R(2 − 2) as in (9), and replace yk by its image in πckρG EP+ ∧ R(2 − 2) . We k (G) will study the smash product of EP+ with the slice tower for R(2 − 2) . The following is [HHR, Lemma 7.14].

Lemma 8.7. For 0

G m k (G) πckρG EP+ ∧ Pm R(2 − 2) = 0 and there is an exact sequence

G k (G) G k (G) πckρG EP+ ∧ PckgR(2 − 2) / πckρG EP+ ∧ R(2 − 2)

G πckρG EP+ ∧ HZ(2) = Z/2

60 A solution to the Arf-Kervaire invariant problem

k (G) Proof. The smash product of EP+ with the slice tower for R(2 − 2) is determined by the restriction of the latter to H, which has the expected properties by induction. The result follows from an examination of its equivariant homotopy groups.

The exact sequence of Lemma 8.7 shows that yk has nontrivial image in the group G G k (G) πckρG EP+ ∧ HZ(2) iﬀ it is not in the image of πckρG EP+ ∧ PckgR(2 − 2) . For this we will identify a property of this image not shared by yk. The following is [HHR, Prop. 7.15]. Proposition 8.8. The image of

∗ i G k (G) G k (G) 0 u k (G) πckρG EP+ ∧ PckgR(2 − 2) / πckρG R(2 − 2) / πckgR(2 − 2) is contained in the image of (γ −1), where γ is a generator of G and i0 denotes restriction to the trivial group.

u k (G) The class yk does not have this property because its image in πckgR(2 − 2) is ∗ g i0N2 rck , which generates a summand isomorphic to the sign representation of G. Hence this result implies the Reduction Theorem. For its proof we refer the reader to the last two pages of [HHR, §7].

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Department of Mathematics, University of Virginia, Charlottesville, VA 22904 E-mail address: [email protected]

Department of Mathematics, Harvard University, Cambridge, MA 02138 E-mail address: [email protected]

Department of Mathematics, University of Rochester, Rochester, NY 14627 E-mail address: [email protected]